Convergence analysis of gradient descent for eigenvector computation

Zhiqiang Xu*, Xin Cao, Xin Gao

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

9 Scopus citations

Abstract

We present a novel, simple and systematic convergence analysis of gradient descent for eigenvector computation. As a popular, practical, and provable approach to numerous machine learning problems, gradient descent has found successful applications to eigenvector computation as well. However, surprisingly, it lacks a thorough theoretical analysis for the underlying geodesically non-convex problem. In this work, the convergence of the gradient descent solver for the leading eigenvector computation is shown to be at a global rate O(min{(λ1p )2 log11 }), where ∆p = λp − λp+1 > 0 represents the generalized positive eigengap and always exists without loss of generality with λi being the i-th largest eigenvalue of the given real symmetric matrix and p being the multiplicity of λ1. The rate is linear at O((λ1p )2 log1 ) if (λ1p )2 = O(1), otherwise sub-linear at O(1 ). We also show that the convergence only logarithmically instead of quadratically depends on the initial iterate. Particularly, this is the first time the linear convergence for the case that the conventionally considered eigengap ∆1 = λ1 − λ2 = 0 but the generalized eigengap ∆p satisfies (λ1p )2 = O(1), as well as the logarithmic dependence on the initial iterate are established for the gradient descent solver. We are also the first to leverage for analysis the log principal angle between the iterate and the space of globally optimal solutions. Theoretical properties are verified in experiments.

Original languageEnglish (US)
Title of host publicationProceedings of the 27th International Joint Conference on Artificial Intelligence, IJCAI 2018
EditorsJerome Lang
PublisherInternational Joint Conferences on Artificial Intelligence
Pages2933-2939
Number of pages7
ISBN (Electronic)9780999241127
DOIs
StatePublished - 2018
Event27th International Joint Conference on Artificial Intelligence, IJCAI 2018 - Stockholm, Sweden
Duration: Jul 13 2018Jul 19 2018

Publication series

NameIJCAI International Joint Conference on Artificial Intelligence
Volume2018-July
ISSN (Print)1045-0823

Conference

Conference27th International Joint Conference on Artificial Intelligence, IJCAI 2018
Country/TerritorySweden
CityStockholm
Period07/13/1807/19/18

ASJC Scopus subject areas

  • Artificial Intelligence

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